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Properties

Label 16.0.513...056.8
Degree $16$
Signature $[0, 8]$
Discriminant $5.140\times 10^{19}$
Root discriminant \(17.06\)
Ramified primes $2,3,7$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133)
 
gp: K = bnfinit(y^16 - 4*y^15 + 16*y^14 - 40*y^13 + 75*y^12 - 118*y^11 + 166*y^10 - 96*y^9 - 32*y^8 + 378*y^7 - 850*y^6 + 888*y^5 - 82*y^4 - 338*y^3 + 974*y^2 - 630*y + 133, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133)
 

\( x^{16} - 4 x^{15} + 16 x^{14} - 40 x^{13} + 75 x^{12} - 118 x^{11} + 166 x^{10} - 96 x^{9} - 32 x^{8} + \cdots + 133 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(51399544780206637056\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 7^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(17.06\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}3^{3/4}7^{1/2}\approx 17.058268835716344$
Ramified primes:   \(2\), \(3\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{9}-\frac{2}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{9}-\frac{2}{7}a^{8}+\frac{1}{7}a^{7}-\frac{3}{7}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}$, $\frac{1}{21}a^{12}+\frac{1}{21}a^{10}+\frac{10}{21}a^{9}-\frac{2}{7}a^{8}+\frac{1}{3}a^{6}+\frac{1}{7}a^{5}-\frac{2}{21}a^{4}-\frac{1}{21}a^{3}-\frac{2}{7}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{21}a^{13}+\frac{1}{21}a^{11}+\frac{1}{21}a^{10}+\frac{1}{7}a^{9}+\frac{4}{21}a^{7}-\frac{1}{7}a^{6}+\frac{10}{21}a^{5}-\frac{1}{21}a^{4}+\frac{2}{7}a^{3}-\frac{8}{21}a^{2}+\frac{1}{3}a$, $\frac{1}{1911}a^{14}-\frac{1}{273}a^{13}-\frac{3}{637}a^{12}+\frac{12}{637}a^{11}-\frac{74}{1911}a^{10}+\frac{905}{1911}a^{9}+\frac{673}{1911}a^{8}-\frac{61}{147}a^{7}-\frac{255}{637}a^{6}-\frac{62}{273}a^{5}-\frac{10}{637}a^{4}-\frac{898}{1911}a^{3}+\frac{87}{637}a^{2}-\frac{44}{273}a+\frac{38}{273}$, $\frac{1}{60\!\cdots\!19}a^{15}-\frac{5703188560}{95430007631413}a^{14}-\frac{56928377617316}{60\!\cdots\!19}a^{13}-\frac{30834002456966}{20\!\cdots\!73}a^{12}+\frac{17524518394416}{668010053419891}a^{11}+\frac{223491646739741}{60\!\cdots\!19}a^{10}+\frac{20344380994521}{51385388724607}a^{9}-\frac{730769050902425}{20\!\cdots\!73}a^{8}+\frac{53241779729647}{60\!\cdots\!19}a^{7}+\frac{319365537571468}{858870068682717}a^{6}+\frac{164776794564080}{668010053419891}a^{5}+\frac{895068805811177}{20\!\cdots\!73}a^{4}-\frac{277353937858219}{60\!\cdots\!19}a^{3}+\frac{52280787139676}{286290022894239}a^{2}-\frac{89691014804857}{858870068682717}a+\frac{17107774553060}{122695724097531}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{578677982}{63360528637} a^{15} + \frac{6760389338}{190081585911} a^{14} - \frac{27394244368}{190081585911} a^{13} + \frac{68298266545}{190081585911} a^{12} - \frac{131079577922}{190081585911} a^{11} + \frac{215042047762}{190081585911} a^{10} - \frac{324716234686}{190081585911} a^{9} + \frac{246559333526}{190081585911} a^{8} - \frac{16641398668}{27154512273} a^{7} - \frac{396570950432}{190081585911} a^{6} + \frac{373654627306}{63360528637} a^{5} - \frac{59008987591}{9051504091} a^{4} + \frac{39203334656}{63360528637} a^{3} + \frac{5447445352}{190081585911} a^{2} - \frac{137017984202}{27154512273} a + \frac{65469261755}{27154512273} \)  (order $6$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{33731591956438}{60\!\cdots\!19}a^{15}-\frac{43855514413427}{20\!\cdots\!73}a^{14}+\frac{538753781321830}{60\!\cdots\!19}a^{13}-\frac{455563307344261}{20\!\cdots\!73}a^{12}+\frac{296373252588934}{668010053419891}a^{11}-\frac{45\!\cdots\!50}{60\!\cdots\!19}a^{10}+\frac{766824586206207}{668010053419891}a^{9}-\frac{147558077864011}{154156166173821}a^{8}+\frac{35\!\cdots\!05}{60\!\cdots\!19}a^{7}+\frac{66\!\cdots\!88}{60\!\cdots\!19}a^{6}-\frac{74\!\cdots\!80}{20\!\cdots\!73}a^{5}+\frac{28\!\cdots\!64}{668010053419891}a^{4}-\frac{12\!\cdots\!58}{858870068682717}a^{3}+\frac{11\!\cdots\!71}{20\!\cdots\!73}a^{2}+\frac{31\!\cdots\!11}{858870068682717}a-\frac{128435866538722}{66066928360209}$, $\frac{10754857022251}{60\!\cdots\!19}a^{15}-\frac{4430830586074}{668010053419891}a^{14}+\frac{174844284002320}{60\!\cdots\!19}a^{13}-\frac{146054353376647}{20\!\cdots\!73}a^{12}+\frac{104160167653911}{668010053419891}a^{11}-\frac{16\!\cdots\!39}{60\!\cdots\!19}a^{10}+\frac{911923835719483}{20\!\cdots\!73}a^{9}-\frac{880886826691832}{20\!\cdots\!73}a^{8}+\frac{27\!\cdots\!94}{60\!\cdots\!19}a^{7}+\frac{782568276871684}{60\!\cdots\!19}a^{6}-\frac{582432522492659}{668010053419891}a^{5}+\frac{84683610350183}{51385388724607}a^{4}-\frac{626351351889994}{858870068682717}a^{3}+\frac{30\!\cdots\!84}{20\!\cdots\!73}a^{2}+\frac{968548167890057}{858870068682717}a-\frac{418683381480493}{858870068682717}$, $\frac{36563412264430}{60\!\cdots\!19}a^{15}-\frac{15254769360849}{668010053419891}a^{14}+\frac{573104075716714}{60\!\cdots\!19}a^{13}-\frac{481391488258745}{20\!\cdots\!73}a^{12}+\frac{316920720959506}{668010053419891}a^{11}-\frac{50\!\cdots\!50}{60\!\cdots\!19}a^{10}+\frac{122511346680377}{95430007631413}a^{9}-\frac{173192604284990}{154156166173821}a^{8}+\frac{56\!\cdots\!45}{60\!\cdots\!19}a^{7}+\frac{53\!\cdots\!36}{60\!\cdots\!19}a^{6}-\frac{24\!\cdots\!80}{668010053419891}a^{5}+\frac{85\!\cdots\!66}{20\!\cdots\!73}a^{4}-\frac{12\!\cdots\!86}{60\!\cdots\!19}a^{3}+\frac{45\!\cdots\!11}{20\!\cdots\!73}a^{2}+\frac{33\!\cdots\!87}{858870068682717}a-\frac{104342177725720}{66066928360209}$, $\frac{39543308296972}{60\!\cdots\!19}a^{15}-\frac{47886090324451}{20\!\cdots\!73}a^{14}+\frac{574908615891385}{60\!\cdots\!19}a^{13}-\frac{446316759263465}{20\!\cdots\!73}a^{12}+\frac{781686143548040}{20\!\cdots\!73}a^{11}-\frac{33\!\cdots\!00}{60\!\cdots\!19}a^{10}+\frac{14\!\cdots\!91}{20\!\cdots\!73}a^{9}-\frac{2677101553061}{51385388724607}a^{8}-\frac{46\!\cdots\!19}{60\!\cdots\!19}a^{7}+\frac{17\!\cdots\!66}{60\!\cdots\!19}a^{6}-\frac{10\!\cdots\!71}{20\!\cdots\!73}a^{5}+\frac{29\!\cdots\!88}{668010053419891}a^{4}+\frac{194885780642822}{122695724097531}a^{3}-\frac{65\!\cdots\!99}{20\!\cdots\!73}a^{2}+\frac{60\!\cdots\!76}{858870068682717}a-\frac{208687507760464}{66066928360209}$, $\frac{15365709381062}{60\!\cdots\!19}a^{15}-\frac{23388694466083}{20\!\cdots\!73}a^{14}+\frac{291543939224087}{60\!\cdots\!19}a^{13}-\frac{273751864920470}{20\!\cdots\!73}a^{12}+\frac{200095282161202}{668010053419891}a^{11}-\frac{34\!\cdots\!98}{60\!\cdots\!19}a^{10}+\frac{93590285302067}{95430007631413}a^{9}-\frac{191928314830403}{154156166173821}a^{8}+\frac{83\!\cdots\!23}{60\!\cdots\!19}a^{7}-\frac{53\!\cdots\!38}{60\!\cdots\!19}a^{6}-\frac{11\!\cdots\!03}{20\!\cdots\!73}a^{5}+\frac{14\!\cdots\!03}{668010053419891}a^{4}-\frac{13\!\cdots\!50}{60\!\cdots\!19}a^{3}+\frac{64\!\cdots\!63}{20\!\cdots\!73}a^{2}-\frac{17\!\cdots\!18}{858870068682717}a+\frac{49400793910549}{66066928360209}$, $\frac{70843288372810}{60\!\cdots\!19}a^{15}-\frac{92239932552697}{20\!\cdots\!73}a^{14}+\frac{85777077792340}{462468498521463}a^{13}-\frac{917203876761752}{20\!\cdots\!73}a^{12}+\frac{17\!\cdots\!88}{20\!\cdots\!73}a^{11}-\frac{80\!\cdots\!44}{60\!\cdots\!19}a^{10}+\frac{37\!\cdots\!43}{20\!\cdots\!73}a^{9}-\frac{721852173453394}{668010053419891}a^{8}-\frac{13\!\cdots\!10}{60\!\cdots\!19}a^{7}+\frac{25\!\cdots\!66}{60\!\cdots\!19}a^{6}-\frac{62\!\cdots\!64}{668010053419891}a^{5}+\frac{18\!\cdots\!21}{20\!\cdots\!73}a^{4}-\frac{838473799246183}{60\!\cdots\!19}a^{3}-\frac{24\!\cdots\!42}{668010053419891}a^{2}+\frac{13\!\cdots\!87}{122695724097531}a-\frac{41\!\cdots\!48}{858870068682717}$, $\frac{192780736}{63360528637}a^{15}-\frac{36219217067}{2471060616843}a^{14}+\frac{49724124046}{823686872281}a^{13}-\frac{61821841826}{353008659549}a^{12}+\frac{136503035386}{353008659549}a^{11}-\frac{86756733554}{117669553183}a^{10}+\frac{1045056369789}{823686872281}a^{9}-\frac{3948674120201}{2471060616843}a^{8}+\frac{107296523646}{63360528637}a^{7}-\frac{2480594980826}{2471060616843}a^{6}-\frac{2613620315440}{2471060616843}a^{5}+\frac{2690605869255}{823686872281}a^{4}-\frac{8189614929359}{2471060616843}a^{3}+\frac{7997961236479}{2471060616843}a^{2}-\frac{118551346573}{117669553183}a-\frac{6960840759}{117669553183}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4124.94709562 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4124.94709562 \cdot 2}{6\cdot\sqrt{51399544780206637056}}\cr\approx \mathstrut & 0.465861090216 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-21}) \), 4.0.3024.2, 4.0.1008.2 x2, 4.2.9408.1 x2, 4.0.189.1, 4.0.21168.1, \(\Q(\sqrt{-3}, \sqrt{7})\), 4.0.432.1, 8.0.7169347584.5, 8.0.7169347584.4, 8.0.448084224.2, 8.0.448084224.7, 8.0.796594176.7, 8.0.9144576.1, 8.0.146313216.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.0.9144576.1, 8.0.9144576.2, 8.0.146313216.1, 8.0.7169347584.8, 8.0.146313216.2, 8.0.7169347584.4, 8.0.448084224.2, 8.0.448084224.1
Degree 16 siblings: 16.4.51399544780206637056.3, 16.0.51399544780206637056.2, 16.0.21407557176262656.1, 16.0.200779471797682176.1, 16.0.51399544780206637056.4, 16.0.51399544780206637056.7
Minimal sibling: 8.0.9144576.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $16$$4$$4$$24$
\(3\) Copy content Toggle raw display 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(7\) Copy content Toggle raw display 7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$